Texture baking is grabbing some textures from the wrong areas
I am not very experienced with texture baking, but it sounds like your problem is due to the pieces intersecting each other when the rays are traced.

If I understand correctly, baking a texture basically creates virtual cameras a certain distance away from the mesh and renders from them, onto the object. This is only because you are trying to bake detail from one mesh to another. If the meshes are the same poly count there should be no reason to do this, so that might be an easy way to fix your problem.

However, in your case, Blender is using the face normals to determine the direction of the rays. This could be inaccurate and it can be more precise if you create a cage mesh so that it can draw a ray from the cage mesh through the vertex of the master mesh, until it hits the low poly mesh to determine where detail should be draw.

It is also possible that the UVs overlapped or were messed up when you joined objects.

Sorry that this isn't a direct answer, but I hope I gave you some ideas.